Curvature Invariants and the Geometric Horizon Conjecture in a Binary Black Hole Merger

Abstract

We study curvature invariants in a binary black hole merger. It has been conjectured that one could define a quasi-local and foliation independent black hole horizon by finding the level--$0$ set of a suitable curvature invariant of the Riemann tensor. The conjecture is the geometric horizon conjecture and the associated horizon is the geometric horizon. We study this conjecture by tracing the level--$0$ set of the complex scalar polynomial invariant, $\mathcal{D}$, through a quasi-circular binary black hole merger. We approximate these level--$0$ sets of $\mathcal{D}$ with level--$\epsilon$ sets of $|\mathcal{D}|$ for small $\epsilon$. We locate the local minima of $|\mathcal{D}|$ and find that the positions of these local minima correspond closely to the level--$\epsilon$ sets of $|\mathcal{D}|$ and we also compare with the level--$0$ sets of $\text{Re}(\mathcal{D})$. The analysis provides evidence that the level--$\epsilon$ sets track a unique geometric horizon. By studying the behaviour of the zero sets of $\text{Re}(\mathcal{D})$ and $\text{Im}(\mathcal{D})$ and also by studying the MOTSs and apparent horizons of the initial black holes, we observe that the level--$\epsilon$ set that best approximates the geometric horizon is given by $\epsilon =10^{-3}$.

16 pages, 4 figures; v2. included three right-hand panels to Figure 2 and updated corresponding explanations. Updated introduction and discussion for clarity. Included mention of future work in discussion/conclusion. Made various minor corrections. To appear in General Relativity and Gravitation

References in corpus (11)

• Introduction to dynamical horizons in numerical relativity
• Classification of the Weyl Tensor in Higher Dimensions and Applications
• Spacetimes characterized by their scalar curvature invariants
• Isolated, slowly evolving, and dynamical trapping horizons: geometry and mechanics from surface deformations
• Dynamics of marginally trapped surfaces in a binary black hole merger: Growth and approach to equilibrium
• News from horizons in binary black hole mergers
• The interior of a binary black hole merger
• The existence and stability of marginally trapped surfaces
• Self-intersecting marginally outer trapped surfaces
• Horizons in a binary black hole merger I: Geometry and area increase
• Cartan Invariants and Event Horizon Detection, Extended Version